Theory of Computation


 Baldric Hood
 3 years ago
 Views:
Transcription
1 Theory of Computation For Computer Science & Information Technology By
2 Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, ContextFree Grammar s and PushDown Automata, Regular and ContextFree Languages, Pumping Lemma, Turing Machines and Undecidability. Analysis of GATE Papers Year Percentage of Marks Over All Percentage % : , Copyright reserved. Web:
3 Contents Contents Chapters Page No. #. Introduction/Preliminaries 3 Introduction Relations 2 3 #2. Finite Automata 4 22 Finite Automata 4 0 Construction of DFA from NFA (Sub Set Construction) 0 Epsilon( )Closures 2 3 Eliminating Transitions (Construction of DFA from NFA) 4 5 Assignment 6 8 Assignment Answer keys & Explanations #3. Regular Expression Definitions Languages Associated with Regular Expressions Algebraic Laws for Regular Expressions Converting Regular Expression to Automata ( NFA) Construction of Regular Expression from Finite Automata 28 3 Ordering the Elimination of States 3 33 Finite Automata with Output Regular Grammar MyhillNerode Theorem Pumping Lemma for Regular Languages Finding (in) Distinguishable States Assignment Answer keys & Explanations #4. Context Free Grammar Introduction Context Free Language Leftmost and Rightmost Derivations Ambiguity Simplification of Context Free Grammar 65 Elimination of Useless Symbols Normal Forms : , Copyright reserved. Web: i
4 Pushdown Automata Definition of PDA NPDA Properties of Context Free Language Decision Algorithms for CFL s NonContext Free Language Pumping Lemma for CFL s 89 9 Membership Algorithm for ContextFree Grammar 9 92 Assignment Assignment Answer keys & Explanations Contents #5. Turing Machines 0 34 Introduction 0 03 Modification of Turing Machine TwoPushdown Stack Machine Counter Machine 08 0 Multiple Tracks 0 6 Universal Turing Machine 6 7 Context Sensitive Grammar 7 8 Linear Bounded Automata 8 9 Hierarchy of Formal Languages (Chomsky Hierarchy) 9 20 Undecidability 20 2 Universal Language 2 27 The Classes P & NP Assignment 30 3 Assignment Answer Keys & Explanations Module Test Test Questions 35 4 Answer Keys & Explanations 4 45 Reference Books 46 : , Copyright reserved. Web: ii
5 CHAPTER "Yesterday I dared to struggle. Today I dare to win."..bernadette Devlin Introduction/Preliminaries Learning Objectives After reading this chapter, you will know:. Relations Introduction String: A string is a finite sequence of symbols put together. E.g. bbac the length of this string is '4'. Note: The length of empty string denoted by, is the string consisting of zero symbols. Thus = 0. Alphabet: An alphabet is a finite set of symbols. E.g.: {a, b, 0,, B} Formal language: A formal language is a set of strings made up of symbols taken from some alphabet. E.g.: The language consisting of all strings formed by the alphabet {0, } Note:. The empty set,, is a formal language. The cardinality (size) of this language is zero. 2. The set consisting of empty string, { } is a formal language. The cardinality (size) of this language is one. Set: A set is a collection of objects (members of the set) without repetition. i. Finite Set: A set which contains finite number of elements is said to be finite set. E.g.: {, 2, 3, 4} ii. Countably Infinite Set: Sets that can be placed in onetoone correspondence with the integers are said to be countably infinite or countable or denumerable. E.g.: The set of the finitelength strings from an alphabet are countably infinite, (if : {0, } Then : {0, 0,, 0, 0...} i.e., all possible strings with 0 and'') iii. Uncountable Set: Sets that can't be placed in onetoone correspondence with the integers are said to be uncountable sets. E.g.: The set of real numbers. : , Copyright reserved. Web:
6 Introduction/Preliminaries Relations A (binary) relation is a set of ordered tuples. The first component of each tuple is chosen from a set called the domain and the second component of each pair is chosen from a (possibly different) set called the range. E.g. Let A {, 2, 3, 4} be a set. A relation R, on 'A' can be defined as R: {(, 2), (, 3), (, ), (2, 3)} Properties of Relations We say a relation R on set S is. Reflexive: If ara for all a in S 2. Irreflexive: If ara is false for all a is S 3. Transitive: If arb and brc implies arc 4. Symmetric: If arb implies bra 5. Asymmetric: If arb implies that bra is false 6. Antisymmetric: If arb and bra implies a = b Equivalence Relation: A relation is said to be equivalence relation if it satisfies the following properties.. Reflexive 2. Symmetric 3. Transitive An important property of equivalence relation 'R' on set 'S' is that R partitions 'S' into disjoint nonempty equivalence classes (may be infinite) i.e., S =..., where for each i and j, with i j. = 2. For each 'a' and 'b' in S jarb is true. 3. For each 'a' in i and 'b' in Sj, arb is false. The 's are called equivalence classes. E.g.: The relation 'R' on people defined by prq if and only if 'p' and 'q' were born at the same hour of the same day of some year: The number of equivalence classes are: 24 (no. of hours) 7 (no. of days in a week). Closure of Relations: Suppose P is a set of properties of relations. The Pclosure of a relation R is the smallest relation R that includes all the pairs of R and possesses the properties in P. E.g.: Let R = {(, 2), (2, 3), (3, )} be a relation on set {, 2, 3} i. The reflexivetransitive closure of R is denoted by R* is Added by Reflexivity (, 2), (2, 3), (3, ), (, ), (2, 2), (3, 3), (, 3), (2, ), (3, 2) Added by Transitivity : , Copyright reserved. Web: 2
7 Introduction/Preliminaries ii. The symmetric closure of R is {(, 2), (2, 3), (3, ), (2, ), (3, 2), (, 3)} Term Definition prefix of s A string obtained by removing zero or more trailing symbols of string s; e.g., ban is a prefix of banana. suffix of s A string formed by deleting zero or more of the leading symbols of s; e.g., nana is a suffix of banana. substring of s A string obtained by deleting a prefix and a suffix from s; e.g., nan is a substring of banana. Every prefix and every suffix of s is a substring of s, but not every substring of s is a prefix or a suffix of s. For every string s, both s and are prefixes, suffixes, and substrings of s. proper prefix, suffix, or Any nonempty string x that is, respectively, a prefix, suffix, or substring of s substring of s such that s x. subsequence of s Any string formed by deleting zero or more not necessarily contiguous symbols from s; e.g., baaa is a subsequence of banana. : , Copyright reserved. Web: 3
8 CHAPTER 2 "Shoot for the moon. Even if you miss, you will land among the stars." Les Brown Finite Automata Learning Objectives After reading this chapter, you will know:. Finite Automata 2. Construction of DFA from NFA (Sub Set Construction) 3. Epsilon( )Closures 4. Eliminating Transitions (Construction of DFA from NFA) Finite Automata A finite automaton involves states and transitions among states in response to inputs. They are useful for building several different kinds of software, including the lexical analysis component of a complier and systems for verifying the correctness of circuits or protocols. They also serve as the control unit in many physical systems including: vending machines, elevators, automatic traffic signals, computer microprocessors, and network protocol stacks. Deterministic Finite Automata A DFA captures the basic elements of an abstract machine: it reads in a string, and depending on the input and the way the machine was designed, it outputs either true or false. A DFA is always in one of N states, which we usually name 0 through N. Each state is labeled true or false. The DFA begins in a designated state called the start state. As the input characters are read in one at a time, the DFA changes from one state to another in a prespecified way. The new state is completely determined by the current state and the character just read in. When the input is exhausted, the DFA outputs true or false according to the label of the state it is currently in. A Deterministic finite automaton is represented by a Quintuple (5tuple): (Q,, δ,, F) Where, Q: Finite set of states : Finite set of input symbols called the alphabet. δ: Q X Q (δ is a transition function from Q X to Q) : A start state, one of the states in Q F: A set of final states, such that F Q : , Copyright reserved. Web: 4
9 Induction Finite Automata Suppose is a string of the form a that is a is the last symbol of, and is the string consisting of all but the last symbol. For example is broken into and a. Then δ (, ) δ(δ (, ), a) Now, to compute δ (, ) first compute δ (, )the state that the automaton is in after processing all but the last symbol of. Suppose this state is p; that is, δ (, ). Then δ (, )is what we get by making a transition from state p on input a. That is, δ (, ) = δ (, a) E.g.: Let us design a DFA to accept the language L = { has both an even number of 0's and an even number of 's} It should not be surprising that the job of the states of this DFA is to count both the number of 0's and the number of 's, but count them modulo 2. That is, the state is used to remember whether the number of 0's seen so far is even or odd, and also to remember whether the number of 's seen so far is even or odd. There are thus four states, which can be given the following interpretations: : Both the number of 0's seen so far and the number of 's seen so far are even. : The number of 0's seen so far is even, but the number of 's seen so far is odd. : The number of 's seen so far is even, but the number of 0's seen so far is odd. : Both the number of 0's seen so far and the number of 's seen so far are odd. State is both the start state and the alone the accepting state. It is the start state, because before reading any inputs, the numbers of 0's and 's seen so far are both zero, and zero is even. It is the only accepting state, because it describes exactly the condition for a sequence of 0's and 's to be in language L. We now know almost how to specify the DFA for language L. It is, A ({,,, }, {, }, δ,, { }) Deterministic Finite Automata Start Transition Diagram for the DFA Where the transition function S is described by the transition diagram notice that, how each input 0 causes the state to cross the horizontal, dashed line. Thus, after seeing an even number of 0's we are always above the line, in state or while after seeing an odd number of 0's we are always below the line, in state or. Likewise, every causes the state to cross the vertical, dashed line. Thus, after seeing an even number of 's. We are always to the left, in state or. While after : , Copyright reserved. Web: 5
10 Finite Automata seeing an odd number of 's we are to the right, in state or. These observations are an informal proof that the four states have the interpretations attributed to them. However, one could prove the correctness of our claims about the states formally, by a mutual induction with respect to example. We can also represent this DFA by a transition table shown below. However, we are not just concerned with the design of this DFA; we want to use it to illustrate the construction of δ from its transition function δ. Suppose the input is 00. Since this string has even numbers of 0's and 's both, we expect it is in the language. Thus, we expect that δ (, 00) = since is the only accepting state. Let us now verify that claim. Transition Table for the DFA The check involves computing δ (, w) for each prefix w of 00, starting at and going in increasing size. The summary of this calculation is: δ (, ) δ (, ) δ(δ (, ), ) δ(, ) δ (, ) δ(δ (, ), ) δ(, ) δ (, ) δ(δ (, ), ) δ(, ) δ (, ) δ(δ (, ), ) δ(, ) δ (, ) δ(δ (, ), ) δ(, ) δ (, ) δ(δ (, ), ) δ(, ) Acceptance by an Automata A string is said to be acce ted by a finite automaton M (Q,, δ,, F) if δ (, x) = for some in F. The language accepted by M, designated L (M), is the set {x δ (, x) is in F}. A language is a regular set (or just regular) if it is the set accepted by some automaton. There are two preferred notations for describing Automata. Transition diagram 2. Transition table : , Copyright reserved. Web: 6
CT32 COMPUTER NETWORKS DEC 2015
Q.2 a. Using the principle of mathematical induction, prove that (10 (2n1) +1) is divisible by 11 for all n N (8) Let P(n): (10 (2n1) +1) is divisible by 11 For n = 1, the given expression becomes (10
More informationCS402  Theory of Automata Glossary By
CS402  Theory of Automata Glossary By Acyclic Graph : A directed graph is said to be acyclic if it contains no cycles. Algorithm : A detailed and unambiguous sequence of instructions that describes how
More informationFinite Automata. Dr. Nadeem Akhtar. Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur
Finite Automata Dr. Nadeem Akhtar Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur PhD Laboratory IRISAUBS University of South Brittany European University
More informationJNTUWORLD. Code No: R
Code No: R09220504 R09 SET1 B.Tech II Year  II Semester Examinations, AprilMay, 2012 FORMAL LANGUAGES AND AUTOMATA THEORY (Computer Science and Engineering) Time: 3 hours Max. Marks: 75 Answer any five
More information(Refer Slide Time: 0:19)
Theory of Computation. Professor somenath Biswas. Department of Computer Science & Engineering. Indian Institute of Technology, Kanpur. Lecture15. Decision Problems for Regular Languages. (Refer Slide
More informationSlides for Faculty Oxford University Press All rights reserved.
Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,
More informationFinal Course Review. Reading: Chapters 19
Final Course Review Reading: Chapters 19 1 Objectives Introduce concepts in automata theory and theory of computation Identify different formal language classes and their relationships Design grammars
More informationCS5371 Theory of Computation. Lecture 8: Automata Theory VI (PDA, PDA = CFG)
CS5371 Theory of Computation Lecture 8: Automata Theory VI (PDA, PDA = CFG) Objectives Introduce Pushdown Automaton (PDA) Show that PDA = CFG In terms of descriptive power Pushdown Automaton (PDA) Roughly
More information6 NFA and Regular Expressions
Formal Language and Automata Theory: CS21004 6 NFA and Regular Expressions 6.1 Nondeterministic Finite Automata A nondeterministic finite automata (NFA) is a 5tuple where 1. is a finite set of states
More information1. [5 points each] True or False. If the question is currently open, write O or Open.
University of Nevada, Las Vegas Computer Science 456/656 Spring 2018 Practice for the Final on May 9, 2018 The entire examination is 775 points. The real final will be much shorter. Name: No books, notes,
More informationVALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203. DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year & Semester : III Year, V Semester Section : CSE  1 & 2 Subject Code : CS6503 Subject
More informationComplexity Theory. Compiled By : Hari Prasad Pokhrel Page 1 of 20. ioenotes.edu.np
Chapter 1: Introduction Introduction Purpose of the Theory of Computation: Develop formal mathematical models of computation that reflect realworld computers. Nowadays, the Theory of Computation can be
More informationUNIT I PART A PART B
OXFORD ENGINEERING COLLEGE (NAAC ACCREDITED WITH B GRADE) DEPARTMENT OF COMPUTER SCIENCE & ENGINEERING LIST OF QUESTIONS YEAR/SEM: III/V STAFF NAME: Dr. Sangeetha Senthilkumar SUB.CODE: CS6503 SUB.NAME:
More informationR10 SET a) Construct a DFA that accepts an identifier of a C programming language. b) Differentiate between NFA and DFA?
R1 SET  1 1. a) Construct a DFA that accepts an identifier of a C programming language. b) Differentiate between NFA and DFA? 2. a) Design a DFA that accepts the language over = {, 1} of all strings that
More informationSkyup's Media. PARTB 2) Construct a Mealy machine which is equivalent to the Moore machine given in table.
Code No: XXXXX JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY, HYDERABAD B.Tech II Year I Semester Examinations (Common to CSE and IT) Note: This question paper contains two parts A and B. Part A is compulsory
More informationT.E. (Computer Engineering) (Semester I) Examination, 2013 THEORY OF COMPUTATION (2008 Course)
*4459255* [4459] 255 Seat No. T.E. (Computer Engineering) (Semester I) Examination, 2013 THEY OF COMPUTATION (2008 Course) Time : 3 Hours Max. Marks : 100 Instructions : 1) Answers to the two Sections
More informationLearn Smart and Grow with world
Learn Smart and Grow with world All Department Smart Study Materials Available Smartkalvi.com TABLE OF CONTENTS S.No DATE TOPIC PAGE NO. UNITI FINITE AUTOMATA 1 Introduction 1 2 Basic Mathematical Notation
More informationTHEORY OF COMPUTATION
THEORY OF COMPUTATION UNIT1 INTRODUCTION Overview This chapter begins with an overview of those areas in the theory of computation that are basic foundation of learning TOC. This unit covers the introduction
More information9.5 Equivalence Relations
9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same
More information(a) R=01[((10)*+111)*+0]*1 (b) ((01+10)*00)*. [8+8] 4. (a) Find the left most and right most derivations for the word abba in the grammar
Code No: R05310501 Set No. 1 III B.Tech I Semester Regular Examinations, November 2008 FORMAL LANGUAGES AND AUTOMATA THEORY (Computer Science & Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE
More informationTOPIC PAGE NO. UNITI FINITE AUTOMATA
TABLE OF CONTENTS SNo DATE TOPIC PAGE NO UNITI FINITE AUTOMATA 1 Introduction 1 2 Basic Mathematical Notation Techniques 3 3 Finite State systems 4 4 Basic Definitions 6 5 Finite Automaton 7 6 DFA NDFA
More informationDerivations of a CFG. MACM 300 Formal Languages and Automata. Contextfree Grammars. Derivations and parse trees
Derivations of a CFG MACM 300 Formal Languages and Automata Anoop Sarkar http://www.cs.sfu.ca/~anoop strings grow on trees strings grow on Noun strings grow Object strings Verb Object Noun Verb Object
More informationLast lecture CMSC330. This lecture. Finite Automata: States. Finite Automata. Implementing Regular Expressions. Languages. Regular expressions
Last lecture CMSC330 Finite Automata Languages Sets of strings Operations on languages Regular expressions Constants Operators Precedence 1 2 Finite automata States Transitions Examples Types This lecture
More informationDecision Properties for Contextfree Languages
Previously: Decision Properties for Contextfree Languages CMPU 240 Language Theory and Computation Fall 2018 Contextfree languages Pumping Lemma for CFLs Closure properties for CFLs Today: Assignment
More informationLECTURE NOTES THEORY OF COMPUTATION
LECTURE NOTES ON THEORY OF COMPUTATION P Anjaiah Assistant Professor Ms. B Ramyasree Assistant Professor Ms. E Umashankari Assistant Professor Ms. A Jayanthi Assistant Professor INSTITUTE OF AERONAUTICAL
More informationECS 120 Lesson 16 Turing Machines, Pt. 2
ECS 120 Lesson 16 Turing Machines, Pt. 2 Oliver Kreylos Friday, May 4th, 2001 In the last lesson, we looked at Turing Machines, their differences to finite state machines and pushdown automata, and their
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2016
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2016 Lecture 15 Ana Bove May 23rd 2016 More on Turing machines; Summary of the course. Overview of today s lecture: Recap: PDA, TM Pushdown
More informationReflection in the Chomsky Hierarchy
Reflection in the Chomsky Hierarchy Henk Barendregt Venanzio Capretta Dexter Kozen 1 Introduction We investigate which classes of formal languages in the Chomsky hierarchy are reflexive, that is, contain
More informationCHAPTER 8. Copyright Cengage Learning. All rights reserved.
CHAPTER 8 RELATIONS Copyright Cengage Learning. All rights reserved. SECTION 8.3 Equivalence Relations Copyright Cengage Learning. All rights reserved. The Relation Induced by a Partition 3 The Relation
More informationComputer Sciences Department
1 Reference Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER 3 D E C I D A B I L I T Y 4 Objectives 5 Objectives investigate the power of algorithms to solve problems.
More informationClosure Properties of CFLs; Introducing TMs. CS154 Chris Pollett Apr 9, 2007.
Closure Properties of CFLs; Introducing TMs CS154 Chris Pollett Apr 9, 2007. Outline Closure Properties of Context Free Languages Algorithms for CFLs Introducing Turing Machines Closure Properties of CFL
More informationTheory of Computation Dr. Weiss Extra Practice Exam Solutions
Name: of 7 Theory of Computation Dr. Weiss Extra Practice Exam Solutions Directions: Answer the questions as well as you can. Partial credit will be given, so show your work where appropriate. Try to be
More informationI have read and understand all of the instructions below, and I will obey the Academic Honor Code.
Midterm Exam CS 341451: Foundations of Computer Science II Fall 2014, elearning section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand all
More informationQUESTION BANK. Unit 1. Introduction to Finite Automata
QUESTION BANK Unit 1 Introduction to Finite Automata 1. Obtain DFAs to accept strings of a s and b s having exactly one a.(5m )(JunJul 10) 2. Obtain a DFA to accept strings of a s and b s having even
More informationUniversity of Nevada, Las Vegas Computer Science 456/656 Fall 2016
University of Nevada, Las Vegas Computer Science 456/656 Fall 2016 The entire examination is 925 points. The real final will be much shorter. Name: No books, notes, scratch paper, or calculators. Use pen
More informationTo illustrate what is intended the following are three write ups by students. Diagonalization
General guidelines: You may work with other people, as long as you write up your solution in your own words and understand everything you turn in. Make sure to justify your answers they should be clear
More informationLECTURE NOTES THEORY OF COMPUTATION
LECTURE NOTES ON THEORY OF COMPUTATION Dr. K Rajendra Prasad Professor Ms. N Mamtha Assistant Professor Ms. S Swarajya Lakshmi Assistant Professor Mr. D Abdulla Assistant Professor INSTITUTE OF AERONAUTICAL
More informationAmbiguous Grammars and Compactification
Ambiguous Grammars and Compactification Mridul Aanjaneya Stanford University July 17, 2012 Mridul Aanjaneya Automata Theory 1/ 44 Midterm Review Mathematical Induction and Pigeonhole Principle Finite Automata
More informationAssignment 4 Solutions of graph problems
Assignment 4 Solutions of graph problems 1. Let us assume that G is not a cycle. Consider the maximal path in the graph. Let the end points of the path be denoted as v 1, v k respectively. If either of
More informationMultiple Choice Questions
Techno India Batanagar Computer Science and Engineering Model Questions Subject Name: Formal Language and Automata Theory Subject Code: CS 402 Multiple Choice Questions 1. The basic limitation of an FSM
More informationLimitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
Computer Language Theory Chapter 4: Decidability 1 Limitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
More information1. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which:
P R O B L E M S Finite Autom ata. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which: a) Are a multiple of three in length. b) End with the string
More informationName: Finite Automata
Unit No: I Name: Finite Automata What is TOC? In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation,
More informationRegular Languages (14 points) Solution: Problem 1 (6 points) Minimize the following automaton M. Show that the resulting DFA is minimal.
Regular Languages (14 points) Problem 1 (6 points) inimize the following automaton Show that the resulting DFA is minimal. Solution: We apply the State Reduction by Set Partitioning algorithm (särskiljandealgoritmen)
More informationOutline. Language Hierarchy
Outline Language Hierarchy Definition of Turing Machine TM Variants and Equivalence Decidability Reducibility Language Hierarchy Regular: finite memory CFG/PDA: infinite memory but in stack space TM: infinite
More informationCS525 Winter 2012 \ Class Assignment #2 Preparation
1 CS525 Winter 2012 \ Class Assignment #2 Preparation Ariel Stolerman 2.26) Let be a CFG in Chomsky Normal Form. Following is a proof that for any ( ) of length exactly steps are required for any derivation
More informationDefinition: A contextfree grammar (CFG) is a 4 tuple. variables = nonterminals, terminals, rules = productions,,
CMPSCI 601: Recall From Last Time Lecture 5 Definition: A contextfree grammar (CFG) is a 4 tuple, variables = nonterminals, terminals, rules = productions,,, are all finite. 1 ( ) $ Pumping Lemma for
More informationTAFL 1 (ECS403) Unit V. 5.1 Turing Machine. 5.2 TM as computer of Integer Function
TAFL 1 (ECS403) Unit V 5.1 Turing Machine 5.2 TM as computer of Integer Function 5.2.1 Simulating Turing Machine by Computer 5.2.2 Simulating Computer by Turing Machine 5.3 Universal Turing Machine 5.4
More informationMidterm Exam II CIS 341: Foundations of Computer Science II Spring 2006, day section Prof. Marvin K. Nakayama
Midterm Exam II CIS 341: Foundations of Computer Science II Spring 2006, day section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand all of
More information1. (10 points) Draw the state diagram of the DFA that recognizes the language over Σ = {0, 1}
CSE 5 Homework 2 Due: Monday October 6, 27 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in
More informationTheory Bridge Exam Example Questions Version of June 6, 2008
Theory Bridge Exam Example Questions Version of June 6, 2008 This is a collection of sample theory bridge exam questions. This is just to get some idea of the format of the bridge exam and the level of
More informationQUESTION BANK. Formal Languages and Automata Theory(10CS56)
QUESTION BANK Formal Languages and Automata Theory(10CS56) Chapter 1 1. Define the following terms & explain with examples. i) Grammar ii) Language 2. Mention the difference between DFA, NFA and εnfa.
More informationComputation Engineering Applied Automata Theory and Logic. Ganesh Gopalakrishnan University of Utah. ^J Springer
Computation Engineering Applied Automata Theory and Logic Ganesh Gopalakrishnan University of Utah ^J Springer Foreword Preface XXV XXVII 1 Introduction 1 Computation Science and Computation Engineering
More informationAutomata & languages. A primer on the Theory of Computation. The imitation game (2014) Benedict Cumberbatch Alan Turing ( ) Laurent Vanbever
Automata & languages A primer on the Theory of Computation The imitation game (24) Benedict Cumberbatch Alan Turing (92954) Laurent Vanbever www.vanbever.eu ETH Zürich (DITET) September, 2 27 Brief CV
More informationRegular Languages and Regular Expressions
Regular Languages and Regular Expressions According to our definition, a language is regular if there exists a finite state automaton that accepts it. Therefore every regular language can be described
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 11 Ana Bove April 26th 2018 Recap: Regular Languages Decision properties of RL: Is it empty? Does it contain this word? Contains
More informationCS154 Midterm Examination. May 4, 2010, 2:153:30PM
CS154 Midterm Examination May 4, 2010, 2:153:30PM Directions: Answer all 7 questions on this paper. The exam is open book and open notes. Any materials may be used. Name: I acknowledge and accept the
More informationDecidable Problems. We examine the problems for which there is an algorithm.
Decidable Problems We examine the problems for which there is an algorithm. Decidable Problems A problem asks a yes/no question about some input. The problem is decidable if there is a program that always
More informationMaterial from Recitation 1
Material from Recitation 1 Darcey Riley Frank Ferraro January 18, 2011 1 Introduction In CSC 280 we will be formalizing computation, i.e. we will be creating precise mathematical models for describing
More informationActually talking about Turing machines this time
Actually talking about Turing machines this time 10/25/17 (Using slides adapted from the book) Administrivia HW due now (Pumping lemma for contextfree languages) HW due Friday (Building TMs) Exam 2 out
More informationFormal Definition of Computation. Formal Definition of Computation p.1/28
Formal Definition of Computation Formal Definition of Computation p.1/28 Computation model The model of computation considered so far is the work performed by a finite automaton Formal Definition of Computation
More informationAUBER (Models of Computation, Languages and Automata) EXERCISES
AUBER (Models of Computation, Languages and Automata) EXERCISES Xavier Vera, 2002 Languages and alphabets 1.1 Let be an alphabet, and λ the empty string over. (i) Is λ in? (ii) Is it true that λλλ=λ? Is
More informationIntroduction to Automata Theory. BİL405  Automata Theory and Formal Languages 1
Introduction to Automata Theory BİL405  Automata Theory and Formal Languages 1 Automata, Computability and Complexity Automata, Computability and Complexity are linked by the question: What are the fundamental
More informationInterpreter. Scanner. Parser. Tree Walker. read. request token. send token. send AST I/O. Console
Scanning 1 read Interpreter Scanner request token Parser send token Console I/O send AST Tree Walker 2 Scanner This process is known as: Scanning, lexing (lexical analysis), and tokenizing This is the
More informationTheory of Programming Languages COMP360
Theory of Programming Languages COMP360 Sometimes it is the people no one imagines anything of, who do the things that no one can imagine Alan Turing What can be computed? Before people even built computers,
More informationProblems, Languages, Machines, Computability, Complexity
CS311 Computational Structures Problems, Languages, Machines, Computability, Complexity Lecture 1 Andrew P. Black Andrew Tolmach 1 The Geography Game Could you write a computer program to play the geography
More informationTheory of Computations Spring 2016 Practice Final Exam Solutions
1 of 8 Theory of Computations Spring 2016 Practice Final Exam Solutions Name: Directions: Answer the questions as well as you can. Partial credit will be given, so show your work where appropriate. Try
More informationONESTACK AUTOMATA AS ACCEPTORS OF CONTEXTFREE LANGUAGES *
ONESTACK AUTOMATA AS ACCEPTORS OF CONTEXTFREE LANGUAGES * Pradip Peter Dey, Mohammad Amin, Bhaskar Raj Sinha and Alireza Farahani National University 3678 Aero Court San Diego, CA 92123 {pdey, mamin,
More informationLanguages and Compilers
Principles of Software Engineering and Operational Systems Languages and Compilers SDAGE: Level I 201213 3. Formal Languages, Grammars and Automata Dr Valery Adzhiev vadzhiev@bournemouth.ac.uk Office:
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105ab/ Today's learning goals Sipser Ch 3.2, 3.3 Define variants of TMs Enumerators Multitape TMs Nondeterministic TMs
More informationA Note on the Succinctness of Descriptions of Deterministic Languages
INFORMATION AND CONTROL 32, 139145 (1976) A Note on the Succinctness of Descriptions of Deterministic Languages LESLIE G. VALIANT Centre for Computer Studies, University of Leeds, Leeds, United Kingdom
More informationCIT3130: Theory of Computation. Regular languages
ƒ CIT3130: Theory of Computation Regular languages ( M refers to the first edition of Martin and H to IALC by Hopcroft et al.) Definitions of regular expressions and regular languages: A regular expression
More informationTotal No. of Questions : 18] [Total No. of Pages : 02. M.Sc. DEGREE EXAMINATION, DEC First Year COMPUTER SCIENCE.
(DMCS01) Total No. of Questions : 18] [Total No. of Pages : 02 M.Sc. DEGREE EXAMINATION, DEC. 2016 First Year COMPUTER SCIENCE Data Structures Time : 3 Hours Maximum Marks : 70 Section  A (3 x 15 = 45)
More informationSource of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D.
Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman And Introduction to Languages and The by J. C. Martin Basic Mathematical
More informationNormal Forms for CFG s. Eliminating Useless Variables Removing Epsilon Removing Unit Productions Chomsky Normal Form
Normal Forms for CFG s Eliminating Useless Variables Removing Epsilon Removing Unit Productions Chomsky Normal Form 1 Variables That Derive Nothing Consider: S > AB, A > aa a, B > AB Although A derives
More informationCS402 Theory of Automata Solved Subjective From Midterm Papers. MIDTERM SPRING 2012 CS402 Theory of Automata
Solved Subjective From Midterm Papers Dec 07,2012 MC100401285 Moaaz.pk@gmail.com Mc100401285@gmail.com PSMD01 MIDTERM SPRING 2012 Q. Point of Kleen Theory. Answer: (Page 25) 1. If a language can be accepted
More informationDefinition 2.8: A CFG is in Chomsky normal form if every rule. only appear on the lefthand side, we allow the rule S ǫ.
CS533 Class 02b: 1 c P. Heeman, 2017 CNF Pushdown Automata Definition Equivalence Overview CS533 Class 02b: 2 c P. Heeman, 2017 Chomsky Normal Form Definition 2.8: A CFG is in Chomsky normal form if every
More informationMIT Specifying Languages with Regular Expressions and ContextFree Grammars. Martin Rinard Massachusetts Institute of Technology
MIT 6.035 Specifying Languages with Regular essions and ContextFree Grammars Martin Rinard Massachusetts Institute of Technology Language Definition Problem How to precisely define language Layered structure
More informationMIT Specifying Languages with Regular Expressions and ContextFree Grammars
MIT 6.035 Specifying Languages with Regular essions and ContextFree Grammars Martin Rinard Laboratory for Computer Science Massachusetts Institute of Technology Language Definition Problem How to precisely
More informationPS3  Comments. Describe precisely the language accepted by this nondeterministic PDA.
University of Virginia  cs3102: Theory of Computation Spring 2010 PS3  Comments Average: 46.6 (full credit for each question is 55 points) Problem 1: Mystery Language. (Average 8.5 / 10) In Class 7,
More informationDHANALAKSHMI SRINIVASAN INSTITUTE OF RESEARCH AND TECHNOLOGY SIRUVACHUR, PERAMBALUR DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
DHANALAKSHMI SRINIVASAN INSTITUTE OF RESEARCH AND TECHNOLOGY SIRUVACHUR, PERAMBALUR621113 DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Third Year CSE( Sem:V) CS2303 THEORY OF COMPUTATION PART B16
More informationA set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N
Mathematical Preliminaries Read pages 529540 1. Set Theory 1.1 What is a set? A set is a collection of entities of any kind. It can be finite or infinite. A = {a, b, c} N = {1, 2, 3, } An entity is an
More informationIntroduction to Computers & Programming
16.070 Introduction to Computers & Programming Theory of computation 5: Reducibility, Turing machines Prof. Kristina Lundqvist Dept. of Aero/Astro, MIT States and transition function State control A finite
More informationChapter 14: Pushdown Automata
Chapter 14: Pushdown Automata Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu The corresponding textbook chapter should
More informationCIS 1.5 Course Objectives. a. Understand the concept of a program (i.e., a computer following a series of instructions)
By the end of this course, students should CIS 1.5 Course Objectives a. Understand the concept of a program (i.e., a computer following a series of instructions) b. Understand the concept of a variable
More informationCMPSCI 250: Introduction to Computation. Lecture 20: Deterministic and Nondeterministic Finite Automata David Mix Barrington 16 April 2013
CMPSCI 250: Introduction to Computation Lecture 20: Deterministic and Nondeterministic Finite Automata David Mix Barrington 16 April 2013 Deterministic and Nondeterministic Finite Automata Deterministic
More information1 Finite Representations of Languages
1 Finite Representations of Languages Languages may be infinite sets of strings. We need a finite notation for them. There are at least four ways to do this: 1. Language generators. The language can be
More informationContext Free Grammars. CS154 Chris Pollett Mar 1, 2006.
Context Free Grammars CS154 Chris Pollett Mar 1, 2006. Outline Formal Definition Ambiguity Chomsky Normal Form Formal Definitions A context free grammar is a 4tuple (V, Σ, R, S) where 1. V is a finite
More informationCS6160 Theory of Computation Problem Set 2 Department of Computer Science, University of Virginia
CS6160 Theory of Computation Problem Set 2 Department of Computer Science, University of Virginia Gabriel Robins Please start solving these problems immediately, and work in study groups. Please prove
More informationFront End: Lexical Analysis. The Structure of a Compiler
Front End: Lexical Analysis The Structure of a Compiler Constructing a Lexical Analyser By hand: Identify lexemes in input and return tokens Automatically: LexicalAnalyser generator We will learn about
More informationCS/B.Tech/CSE/IT/EVEN/SEM4/CS402/ ItIauIafIaAblll~AladUnrtel1ity
CS/B.Tech/CSE/IT/EVEN/SEM4/CS402/201516 ItIauIafIaAblll~AladUnrtel1ity ~ t; ~~ ) MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY, WEST BENGAL Paper Code: CS402 FORMAL LANGUAGE AND AUTOMATA THEORY
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20ab/ Final exam The final exam is Saturday December 16 11:30am2:30pm. Lecture A will take the exam in Lecture B will take the exam
More informationTheory of Computations Spring 2016 Practice Final
1 of 6 Theory of Computations Spring 2016 Practice Final 1. True/False questions: For each part, circle either True or False. (23 points: 1 points each) a. A TM can compute anything a desktop PC can, although
More informationUniversal Turing Machine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice s Theorem Decidability Continued
CD5080 AUBER odels of Computation, anguages and Automata ecture 14 älardalen University Content Universal Turing achine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice s Decidability
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105ab/ Today's learning goals Sipser Ch 1.2, 1.3 Decide whether or not a string is described by a given regular expression
More information(DMTCS 01) Answer Question No.1 is compulsory (15) Answer One question from each unit (4 15=60) 1) a) State whether the following is True/False:
(DMTCS 01) M.Tech. DEGREE EXAMINATION, DECEMBER  2015 (Examination at the end of First Year) COMPUTER SCIENCE Paper  I : Data structures Time : 03 Hours Maximum Marks : 75 Answer Question No.1 is compulsory
More informationTheory of Computation, Homework 3 Sample Solution
Theory of Computation, Homework 3 Sample Solution 3.8 b.) The following machine M will do: M = "On input string : 1. Scan the tape and mark the first 1 which has not been marked. If no unmarked 1 is found,
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20ab/ Final exam The final exam is Saturday March 18 8am11am. Lecture A will take the exam in GH 242 Lecture B will take the exam
More information10/11/2018. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings. Partial Orderings
Sometimes, relations define an order on the elements in a set. Definition: A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set
More informationA Characterization of the Chomsky Hierarchy by String Turing Machines
A Characterization of the Chomsky Hierarchy by String Turing Machines Hans W. Lang University of Applied Sciences, Flensburg, Germany Abstract A string Turing machine is a variant of a Turing machine designed
More information